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152 Denault and PelletierSee, for example, the section of ref. 17 on the distribution of balls in urns, for thedetails on Eq. 15.Because the draws follow a multinomial distribution, the probability p iof occurrenceof decapeptide i with associated 5-tuple (n 1,n 2,n 3,n 4,n 5), is:pin1 n2= p n , n , n , n , n = ⎛ ⎝ ⎜ 1 ⎞⎠⎟ ⎛ ⎝ ⎜ 2 ⎞(1 2 3 4 5)64 64⎠⎟n3 n4 n5⎛ 3 ⎞ 4 6⎝⎜64⎠⎟ ⎛ ⎝ ⎜ ⎞64⎠⎟ ⎛ ⎝ ⎜ ⎞64⎠⎟(16)so that one can easily compute λ i=λ(n 1,n 2,n 3,n 4,n 5) through Eq. 14.21We can finally compute λ = ∑ 10iλ using only the 1001 values for λ(n 1,n 2,n 3,n 4,n 5)= 1 iand the number of times each is repeated, given by Eq. 13:λ ===≅2110∑i=1λi∑⎡ 10!219 22 35 435⎢n n n n n ⎤1 2 3 4 5⎣nnnn1! 2! 3!! !⎥nnnnn , , , ,n⎦λ( )( n1, n2, n3, n4,n5) 4 5∑( n1, n2, n3, n4,n5)∑(n1, n2, n3, n4,n5)⎡ 10!n n2192 n3 n4 n ⎤5⎢2 5 3 1 pn1 n2 n3 n4 n5⎣nnnnn1! 2! 3! 4! 5!⎥⎡⎣ − ( , , , , ) ⎤ ⎦⎦⎡ 10!n1 n2n2 9 23 n4 n ⎤5 p n1 n2 n3 n4 n5m⎢5 3 e⎣nnnnn1! 2! 3! 4! 5!⎥ – ( , , , , )⎦where the second, third, and fourth summations are over all 5-tuples (n 1,n 2,n 3,n 4,n 5)5defined in Heading 1. That is, they are sets of five numbers such that ∑ n ,i i= 10= 1where the n iare numbers in the set {0,1,…,10} because it is a decapeptide. Also,p(n 1,n 2,n 3,n 4,n 5) is computed according to Eq. 16. The last equation gives an approximationof λ (justified by the fact that 1+ x ≅e x for small x), which is much easier tohandle than the second to last equation; too often, 1–p(n 1,n 2,n 3,n 4,n 5)would round upto 1 on a computer.8. See Note 5. The only difference here is that the binomial distribution’s underlyingprobability is not 1/n (the probability in the equiprobable context), but some otherprobability p, which depends on the variant.9. Recall the meaning of the level of significance α =0.05. It means that there is a5% chance of a type I error, which is to reject the hypothesis although it is true.That is, there is a 1:20 chance of declaring “flawed” a perfectly well-executedexperiment.AcknowledgmentsWe thank Andreas Plückthun and group members for proposing problems aswell as Bruno Rémillard (HEC Montréal, Canada), Nicolas Doucet (Université dem